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MCDM Methods

Step-by-Step Algorithms

Spherical Fuzzy TOPSIS Method

Spherical Fuzzy TOPSIS Method

Video Demonstration

Step-by-Step Algorithm

1

Form the fuzzy decision matrix

Decision maker \(k\) evaluates alternative \(A_i\) with respect to criterion \(C_j\) using a spherical fuzzy number :

\[ \widetilde{x}_{ij}^{(k)}=\left( \mu_{ij}^{(k)}, \nu_{ij}^{(k)}, \pi_{ij}^{(k)} \right) \]

\(\mu\), \(\nu\), and \(\pi\) represent membership, non-membership, and hesitancy degrees, respectively.

2

Aggregated spherical fuzzy decision matrix

Formula

Aggregate the judgments of each decision maker (DM) and the criteria weights using Spherical Weighted Arithmetic Mean (SWAM) or Spherical Weighted Geometric Mean (SWGM) operators.

Each decision-maker \(DM_k\) may have a different importance weight \(w_k\). The weights represent the relative importance of the decision-makers in the group decision-making process.

\[ w_k \geq 0 \]
\[ \sum_{k=1}^{K} w_k=1 \]

\[ \widetilde{x}_{ij}=\operatorname{SWAM} \left( \widetilde{x}_{ij}^{(1)}, \widetilde{x}_{ij}^{(2)}, \ldots, \widetilde{x}_{ij}^{(K)} \right) \]

\[ \operatorname{SWAM} \left( \widetilde{A}_{S1}, \widetilde{A}_{S2}, \ldots, \widetilde{A}_{SK} \right)=\left\{ \begin{array}{c} \left[ 1- \displaystyle\prod_{i=1}^{n} \left(1-\mu_{\widetilde{A}_{Si}}^{\,2}\right)^{w_i} \right]^{\frac{1}{2}} \ ,\ \\[12pt] \displaystyle \prod_{i=1}^{n} \left(v_{\widetilde{A}_{Si}}\right)^{w_i} \ ,\ \\[12pt] \left[ \displaystyle\prod_{i=1}^{n} \left(1-\mu_{\widetilde{A}_{Si}}^{\,2}\right)^{w_i} - \displaystyle\prod_{i=1}^{n} \left( 1-\mu_{\widetilde{A}_{Si}}^{\,2} -\pi_{\widetilde{A}_{Si}}^{\,2} \right)^{w_i} \right]^{\frac{1}{2}} \end{array} \right\} \]

\[ \widetilde{x}_{ij}=\operatorname{SWGM} \left( \widetilde{x}_{ij}^{(1)}, \widetilde{x}_{ij}^{(2)}, \ldots, \widetilde{x}_{ij}^{(K)} \right) \]

\[ \operatorname{SWGM} \left( \widetilde{A}_{S1}, \widetilde{A}_{S2}, \ldots, \widetilde{A}_{SK} \right)=\left\{ \begin{array}{c} \displaystyle \prod_{i=1}^{n} \left(\mu_{\widetilde{A}_{Si}}\right)^{w_i} \ ,\ \\[12pt] \left[ 1- \displaystyle\prod_{i=1}^{n} \left(1-\nu_{\widetilde{A}_{Si}}^{\,2}\right)^{w_i} \right]^{\frac{1}{2}} \ ,\ \\[12pt] \left[ \displaystyle\prod_{i=1}^{n} \left(1-\nu_{\widetilde{A}_{Si}}^{\,2}\right)^{w_i} - \displaystyle\prod_{i=1}^{n} \left( 1-\nu_{\widetilde{A}_{Si}}^{\,2} -\pi_{\widetilde{A}_{Si}}^{\,2} \right)^{w_i} \right]^{\frac{1}{2}} \end{array} \right\} \]
3

Construct the aggregated weighted spherical fuzzy decision matrix

Formula

The aggregated spherical fuzzy decision value is combined with the criterion weight :

\[ \widetilde{v}_{ij}=\widetilde{x}_{ij} \otimes \widetilde{w}_j \]
\[ \widetilde{x}_{ij} \otimes \widetilde{w}_j=\left\{ \begin{array}{c} \mu_{\widetilde{x}_{ij}} \mu_{\widetilde{w}_j} \ ,\ \\[10pt] \left[ \nu_{\widetilde{x}_{ij}}^{\,2} + \nu_{\widetilde{w}_j}^{\,2} - \nu_{\widetilde{x}_{ij}}^{\,2} \nu_{\widetilde{w}_j}^{\,2} \right]^{\frac{1}{2}} \ ,\ \\[10pt] \left[ \left(1-\nu_{\widetilde{w}_j}^{\,2}\right) \pi_{\widetilde{x}_{ij}}^{\,2} + \left(1-\nu_{\widetilde{x}_{ij}}^{\,2}\right) \pi_{\widetilde{w}_j}^{\,2} - \pi_{\widetilde{x}_{ij}}^{\,2} \pi_{\widetilde{w}_j}^{\,2} \right]^{\frac{1}{2}} \end{array} \right\} \]
\[ \widetilde{w}_j=\left( \mu_{w_j}, \nu_{w_j}, \pi_{w_j} \right) \]

Therefore, the aggregated weighted spherical fuzzy decision matrix is :

\[ \widetilde{V}=[\widetilde{v}_{ij}]_{m\times n} \]
4

Score function values

Formula

Defuzzify the aggregated weighted decision matrix by using the proposed score function.

\[ \operatorname{Score}(\widetilde{v}_{ij})=\left( \mu_{\widetilde{v}_{ij}} - \pi_{\widetilde{v}_{ij}} \right)^2 - \left( \nu_{\widetilde{v}_{ij}} - \pi_{\widetilde{v}_{ij}} \right)^2 \]
5

Spherical fuzzy positive ideal solution and spherical fuzzy negative ideal solution

Formula

Determine the SFPIS and SFNIS for each criterion based on the score values of the aggregated weighted spherical fuzzy decision matrix.

\[ \widetilde{v}_{j}^{+}=\widetilde{v}_{ij} \quad\text{corresponding to}\quad \max_i Score\left(\widetilde{v}_{ij}\right) \]
\[ \widetilde{v}_{j}^{-}=\widetilde{v}_{ij} \quad\text{corresponding to}\quad \min_i Score\left(\widetilde{v}_{ij}\right) \]
\[ X^{+}=\left\{ \widetilde{v}_{1}^{+}, \widetilde{v}_{2}^{+}, \ldots, \widetilde{v}_{n}^{+} \right\}=\left\{ C_1,\left(\mu_1^{+},\nu_1^{+},\pi_1^{+}\right), C_2,\left(\mu_2^{+},\nu_2^{+},\pi_2^{+}\right), \ldots, C_n,\left(\mu_n^{+},\nu_n^{+},\pi_n^{+}\right) \right\} \]
\[ X^{-}=\left\{ \widetilde{v}_{1}^{-}, \widetilde{v}_{2}^{-}, \ldots, \widetilde{v}_{n}^{-} \right\}=\left\{ C_1,\left(\mu_1^{-},\nu_1^{-},\pi_1^{-}\right), C_2,\left(\mu_2^{-},\nu_2^{-},\pi_2^{-}\right), \ldots, C_n,\left(\mu_n^{-},\nu_n^{-},\pi_n^{-}\right) \right\} \]
6

Distances to positive and negative ideal solutions, closeness ratio and ranking

Formula

Calculate the distance of each alternative from the spherical fuzzy positive ideal solution and the spherical fuzzy negative ideal solution.

Distance to SFPIS :

\[ D_i^{+}=\sqrt{ \frac{1}{2n} \sum_{i=1}^{n} \left[ \left( \mu_{\widetilde{v}_{ij}} - \mu_{\widetilde{v}_{j}^{+}} \right)^2 + \left( \nu_{\widetilde{v}_{ij}} - \nu_{\widetilde{v}_{j}^{+}} \right)^2 + \left( \pi_{\widetilde{v}_{ij}} - \pi_{\widetilde{v}_{j}^{+}} \right)^2 \right]} \]

Distance to SFNIS :

\[ D_i^{-}=\sqrt{ \frac{1}{2n} \sum_{i=1}^{n} \left[ \left( \mu_{\widetilde{v}_{ij}} - \mu_{\widetilde{v}_{j}^{+}} \right)^2 + \left( \nu_{\widetilde{v}_{ij}} - \nu_{\widetilde{v}_{j}^{+}} \right)^2 + \left( \pi_{\widetilde{v}_{ij}} - \pi_{\widetilde{v}_{j}^{+}} \right)^2 \right]} \]

Calculate the relative closeness of each alternative to the spherical fuzzy positive ideal solution.

\[ \ CC_i=\frac{ D_i^{-}}{ D_i^{+}+D_i^{-}} \]

Rank all alternatives according to their closeness ratios in descending order. A larger \(CC_i\) indicates a better alternative.

\[ CC_{(1)} >CC_{(2)} >\cdots >CC_{(m)} \]

SFTOPSIS Algorithm Flowchart